Exceptional Magnetocaloric Responses in a Gadolinium Silicate with Strongly Correlated Spin Disorder for Sub‐Kelvin Magnetic Cooling

Abstract The development of magnetocaloric materials with a significantly enhanced volumetric cooling capability is highly desirable for the application of adiabatic demagnetization refrigerators in confined spatial environments. Here, the thermodynamic characteristics of a magnetically frustrated spin‐7/2 Gd9.33[SiO4]6O2 is presented, which exhibits strongly correlated spin disorder below ≈1.5 K. A quantitative model is proposed to describe the magnetization results by incorporating nearest‐neighbor Heisenberg antiferromagnetic and dipolar interactions. Remarkably, the recorded magnetocaloric responses are unprecedentedly large and applicable below 1.0 K. It is proposed that the S = 7/2 spin liquids serve as versatile platforms for investigating high‐performance magnetocaloric materials in the sub‐kelvin regime, particularly those exhibiting a superior cooling power per unit volume.


I. Rietveld refinement
Powder X-ray diffraction patterns (PXRD) were collected at room temperature on a Huber diffraction and positioning equipment with monochromatic Cu Kα radiation.Rietveld refinements with the pseudo-Voigt function were performed within the GSAS package (EXPGUI), of which the Chebyshev polynomial of 10 terms was used to model the backgrounds [1,2] .

II. Magnetic modeling
The free Heisenberg spin model that describes the bulk magnetization M, under an applied field of H and temperature T, takes the form 0 () of which the saturation magnetization M0 = JgJμB, x = JgJμBμ0H/kBT, and BJ(x) represents the Brillouin function, where gJ = 2 is the spectroscopic splitting factor for Gd 3+ ions, J = S = 7/2 denotes the electronic spin, μB is the Bohr magneton and kB is the Boltzmann constant.
The dipolar energy scale, Dnn, is estimated by taking the form, where Jex is the exchange constant and Nnn represents the number of nearest neighbors for a single Gd 3+ .We here adopted a crude molecular model to characterize the nn exchange, Jnn, by assuming that an internal field Hi is generated with the partial ordering of spin-7/2 Gd 3+ ions [3][4][5][6] .The total effective field acting on magnetic ions then reads as where Hext and Hd represent the applied field and the dipolar field, Hi is proportional to the degree of order in the system.This approach is justified in the case of the ensembled isotropic Gd 3+ spins (the contribution of orbital momentum L is zero) and in the analyzed paramagnetic regime where quantum fluctuations can be neglected.
Considering the exchange part of the Hamiltonian only, we get

III. Analysis of heat capacity and magnetic entropy
According to the thermodynamic Maxwell relation, the isothermal magnetic entropy change can be related to the magnetization dependences by  We then adopted a mean-field approach, corresponding to the Jnn approximation, to bring the calculated results into closer agreement with experimental values.The model here starts from a paramagnetic phase, where the magnetic entropy Smag is related to the partition function Z by

IV. Scaling analysis
The order of magnetic phase transitions can be determined by assessing the universality of the magnetocaloric responses [7][8][9][10][11] .A single master curve (Figure S5) is constructed by normalizing the −ΔSmag(H, T) datasets phenomenologically with respect to the corresponding maximum by ΔSmag(H, T)/ΔSmax, and rescaling the temperature axis to  [7][8][9][10][11] .A scaling nature of the trend of n(H, T)→2 is identified, as illustrated in Figure S5, which resembles those of the second-order thermomagnetic phase transitions and in accordance with the universal analysis [10,12] .It should be noted that the regions of low applied fields (H < 0.5 T) were omitted for possibilities of 'multi-domain' state [10] .

Figure S1 .
Figure S1.Observed and simulated diffraction patterns for GSO (a) and La9.33 (b), along with the differences in Obs − Cal and the Bragg positions (vertical bars).
effective magnetic moments, μeff = gJ(J 2 +J) 1/2 , Rnn denotes the nn distance determined by the Rietveld refinements.The effective exchange constant can be related to the Weiss temperature ѲW via a standard mean-field estimate, the exchange parameter Jnn.

Figure S2 .
Figure S2.(a) The derivative of 1/χ(T) (close marks), and the corresponding derivative of the Curie-Weiss fit (dash line).(b) The derivative of the field-dependent magnetization, dM/dH, down to 1.3 K. (c) The dc susceptiobility in ther form of χT vs. T.

where
data collected at discrete temperature T and field H intervals.Additionally, the numerical calculation can be processed from heat capacity data, by Smag(T, 0) denotes the zero-field magnetic entropy, and Cmag(T, H) = Cp(T, H) − CL represents magnetic heat capacity.The low-temperature lattice contributions CL (0.05 K < T < 3 K) is determined by fitting and extrapolating the heat capacity data of La9.33 with a single Debye model, NA denotes the Avogadro constant, ΘD denotes the Debye temperature, and n is the number of atoms per chemical formula, respectively.

Figure S3 .
Figure S3.(a) Heat capacity of La9.33 and the single Debye fit.(b) Temperature dependences of magnetic heat capacity Cmag under different magnetic fields.The dashed line represents a two-level Schottky fitting.
the partition function Z takes the form, MJ = gJμBJ represents the magnetic moment of one atom.Summation of Eq. (13-14) gives the magnetic entropy Smag per mole as mag sinh[(2 1) / 2 ] ln ( ) sinh( / 2 )where x = gJμBS(H+Hi)/kBT.The Smag-T phase diagram was thus modified considering the discrepancy between the calculated data and the model results.

Figure S4 .
Figure S4.(a) Magnetic entropy change of some representative ADR materials under external fields of 1 T. (b) Adiabatic temperature change, Tad, under varied constant fields.
Tmax represents the temperature of −ΔSmax, and Tref corresponds to the reference temperature.Experientially, the Tref here was defined by the temperature corresponding to the half maximum of the peak values of the magnetic entropy change curves, ΔSmag(H, Tref) = 1/2ΔSmax

Figure S5 .
Figure S5.(a) The phenomenological universal curves derived from a total of 36 applied fields.(b) The field dependences of the exponent n(H, T) of isothermal curves in the temperature range of 0.4− 11K.

Table S1 .
Crystallographic parameters and the equivalent isotropic displacement parameters of GSO.

Table S2 .
Selective bond lengths and bond angles of GSO.